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IB AI SL Math

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N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>With reference to the origin $O$, the coordinates of $A$, $B$ and $C$ are $(2,8,-a)$, $(1,-10,3)$ and $(5,6,a)$ respectively. The plane $\pi$ contains points $O$, $A$, $B$ and $C$.</p>
<ol class="abc-list"><li data-list-item-id="ri-q11-l2-item01" data-list-item-path="ri-q11-l2-item01"><div class="list-item-paragraph">Show that&nbsp;$a=1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="ri-q11-l2-item02" data-list-item-path="ri-q11-l2-item02"><div class="list-item-paragraph">Find the acute angle between $\pi$ and the $yz$-plane.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="ri-q11-l2-item03" data-list-item-path="ri-q11-l2-item03"><div class="list-item-paragraph">$\pi$ and the $yz$-plane meet in the line $l$. By evaluating an appropriate cross product, find a vector equation for&nbsp;$l$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="ri-q11-l2-item04" data-list-item-path="ri-q11-l2-item04"><div class="list-item-paragraph">Write down the distance from $A$ to the $yz$-plane. Hence or otherwise, find the distance from $A$ to $l$.<span class="inline-marks">[3]</span></div></li></ol>

With reference to the origin $O$, the coordinates of $A$, $B$ and $C$ are $(2,8,-a)$, $(1,-10,3)$ and $(5,6,a)$ respectively. The plane $\pi$ contains points $O$, $A$, $B$ and $C$.

Find the acute angle between $\pi$ and the $yz$-plane.

$\pi$ and the $yz$-plane meet in the line $l$. By evaluating an appropriate cross product, find a vector equation for $l$.

Write down the distance from $A$ to the $yz$-plane. Hence or otherwise, find the distance from $A$ to $l$.

Relationships Between a Point and a Plane

Distance from a Point to a Plane

Equations of Planes

Angle Between Two Planes

Line of Intersection of Two Planes

Distance from a Point to a Line

2026 RI P1 Q11

Vectors

<p>The plane $\Pi_1$ is defined by the equation $\mathbf{r}\cdot((\beta-1)\mathbf{i}+\beta\mathbf{j}+2\beta\mathbf{k})=-3$, where&nbsp;$\beta$ is a real constant. A point $P$ has coordinates $(3,1,-2)$.</p>
<ol class="abc-list"><li data-list-item-id="hci-q08-item01" data-list-item-path="hci-q08-item01"><div class="list-item-paragraph">Verify that the point $P$ lies on $\Pi_1$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="hci-q08-item02" data-list-item-path="hci-q08-item02"><div class="list-item-paragraph">The plane $\Pi_2$ contains $P$ and is parallel to the vectors $\begin{pmatrix}1\\1\\2\end{pmatrix}$ and $\begin{pmatrix}1\\3\\4\end{pmatrix}$.</div><div class="list-item-paragraph">Find, in terms of $\beta$, the vector equation of the line $l$ where&nbsp;$\Pi_1$ and $\Pi_2$ meet.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="hci-q08-item03" data-list-item-path="hci-q08-item03"><div class="list-item-paragraph">The plane $\Pi_3$ has cartesian equation $x+2y=29-4z$.</div><div class="list-item-paragraph">It is known that there is no common point of intersection between $\Pi_1$, $\Pi_2$ and $\Pi_3$. Comment on the relationship between $l$ and $\Pi_3$. Hence find the value of $\beta$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="hci-q08-item04" data-list-item-path="hci-q08-item04"><div class="list-item-paragraph">It is now given that&nbsp;$\beta=3$.</div><div class="list-item-paragraph">Given that a point $A$ lies on $\Pi_3$ and $P$ is the foot of the perpendicular from $A$ to $\Pi_1$, find the coordinates of $A$.<span class="inline-marks">[4]</span></div></li></ol>

The plane $\Pi_1$ is defined by the equation $\mathbf{r}\cdot((\beta-1)\mathbf{i}+\beta\mathbf{j}+2\beta\mathbf{k})=-3$, where $\beta$ is a real constant. A point $P$ has coordinates $(3,1,-2)$.

Verify that the point $P$ lies on $\Pi_1$.

Perpendicular Projection and Reflection in a Plane

Relationships Between a Line and a Plane

Intersection of a Line and a Plane

2026 HCI P1 Q8

<p>The planes $\pi_1$ and $\pi_2$ have equations</p>
<p style="text-align: center">${2x + 2y + z = 2,}$<br>${5x - 2y - z = 5,}$</p>
<p>respectively.</p>
<ol class="i-list"><li data-list-item-id="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-0" data-list-item-path="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-0"><div class="list-item-paragraph">Find the acute angle between $\pi_1$ and $\pi_2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-1" data-list-item-path="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-1"><div class="list-item-paragraph">Verify that the point $P$ with position vector $\mathbf{i}+\mathbf{j}-2\mathbf{k}$ lies in $\pi_1$ and $\pi_2$. Hence, find a vector equation of the line of intersection of $\pi_1$ and $\pi_2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-2" data-list-item-path="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-2"><div class="list-item-paragraph">The point $Q$ has position vector $\alpha\mathbf{i}+2\alpha\mathbf{k}$, where&nbsp;$\alpha \in \mathbb{R}$. Show that the perpendicular distance from $Q$ to $\pi_1$ is $\dfrac{2|2\alpha-1|}{3}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-3" data-list-item-path="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-3"><div class="list-item-paragraph"><p>Hence or otherwise, find the cartesian equations of the two planes that are parallel to $\pi_1$ and at a distance of $6$ units from $\pi_1$.</p><span class="inline-marks">[4]</span></div></li></ol>

The planes $\pi_1$ and $\pi_2$ have equations

${2x + 2y + z = 2,}$
${5x - 2y - z = 5,}$

Find the acute angle between $\pi_1$ and $\pi_2$.

Verify that the point $P$ with position vector $\mathbf{i}+\mathbf{j}-2\mathbf{k}$ lies in $\pi_1$ and $\pi_2$. Hence, find a vector equation of the line of intersection of $\pi_1$ and $\pi_2$.

The point $Q$ has position vector $\alpha\mathbf{i}+2\alpha\mathbf{k}$, where $\alpha \in \mathbb{R}$. Show that the perpendicular distance from $Q$ to $\pi_1$ is $\dfrac{2|2\alpha-1|}{3}$.

2025 SAJC Promo Q10

<p>The plane $p_1$ has equation $x+y+z=3$ while the line $l_1$ passes through the origin and is parallel to $-2\mathbf{i}+3\mathbf{j}-2\mathbf{k}$. It is given that the point $A$ has position vector $\mathbf{i}$.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-1-item-0" data-list-item-path="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-1-item-0"><div class="list-item-paragraph">Find the acute angle, in degrees, between $l_1$ and $p_1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-1-item-1" data-list-item-path="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-1-item-1"><div class="list-item-paragraph">Find the exact coordinates of the foot of perpendicular from the point $A$ to $p_1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-1-item-2" data-list-item-path="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-1-item-2"><div class="list-item-paragraph">Point $B$ is a point on the line $l_1$. It is given that the shortest distance from point $B$ to the plane $p_1$ is $\sqrt{27}$ units, find the possible coordinates of $B$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The plane $p_2$ has equation $3x+4y+\alpha z=7+\alpha$, $\alpha \in \mathbb{R}$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-3-item-0" data-list-item-path="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-3-item-0"><div class="list-item-paragraph">It is given that the point $C$ with coordinates $(1,1,1)$ lies on $p_1$. Show that point $C$ also lies on $p_2$. Hence find a vector equation of the line $l_2$, the line of intersection between $p_1$ and $p_2$, in terms of $\alpha$.<span class="inline-marks">[3]</span></div></li></ol>

The plane $p_1$ has equation $x+y+z=3$ while the line $l_1$ passes through the origin and is parallel to $-2\mathbf{i}+3\mathbf{j}-2\mathbf{k}$. It is given that the point $A$ has position vector $\mathbf{i}$.

Find the acute angle, in degrees, between $l_1$ and $p_1$.

Find the exact coordinates of the foot of perpendicular from the point $A$ to $p_1$.

Point $B$ is a point on the line $l_1$. It is given that the shortest distance from point $B$ to the plane $p_1$ is $\sqrt{27}$ units, find the possible coordinates of $B$.

The plane $p_2$ has equation $3x+4y+\alpha z=7+\alpha$, $\alpha \in \mathbb{R}$.

It is given that the point $C$ with coordinates $(1,1,1)$ lies on $p_1$. Show that point $C$ also lies on $p_2$. Hence find a vector equation of the line $l_2$, the line of intersection between $p_1$ and $p_2$, in terms of $\alpha$.

Angle Between a Line and a Plane

2025 ASRJC Promo Q10

<p>The point $A$ has coordinates $(1, -1, 2)$ and the line $l$ has equation $\mathbf{r} = 2\mathbf{i} - 3\mathbf{j} + \lambda(\mathbf{i} - 2\mathbf{j} + 5\mathbf{k})$. The point $A$ and the line $l$ lies in the plane $p_1$.</p>
<ol class="i-list"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph"><p>Find the position vector of the foot of perpendicular from $A$ to the line $l$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-2-item-1" data-list-item-path="imported-item-block-2-item-1"><div class="list-item-paragraph">Show that the equation of $p_1$ is $2x + y = 1$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The plane $p_2$ with equation $8x + ay + z = 4$ meets the plane $p_1$ at the line $L$ and the point $B(0, 1, 0)$ lies on $L$.</p>
<ol class="i-list" start="3"><li data-list-item-id="imported-item-block-4-item-0" data-list-item-path="imported-item-block-4-item-0"><div class="list-item-paragraph">Show that&nbsp;$L$ is parallel to $\mathbf{i} - 2\mathbf{j} + (2a - 8)\mathbf{k}$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-4-item-1" data-list-item-path="imported-item-block-4-item-1"><div class="list-item-paragraph">Show that&nbsp;$a = 4$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-4-item-2" data-list-item-path="imported-item-block-4-item-2"><div class="list-item-paragraph">Find the angle between $l$ and $p_2$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Another point $C$ lies in the plane $p_1$ such that&nbsp;$BC$ is perpendicular to the line $L$.</p>
<ol class="i-list" start="6"><li data-list-item-id="imported-item-block-6-item-0" data-list-item-path="imported-item-block-6-item-0"><div class="list-item-paragraph">Show that&nbsp;$\overrightarrow{BC}$ is parallel to $\mathbf{k}$. Given that the distance of $C$ from $p_2$ is 5 units, find the possible coordinates of $C$.<span class="inline-marks">[4]</span></div></li></ol>

The point $A$ has coordinates $(1, -1, 2)$ and the line $l$ has equation $\mathbf{r} = 2\mathbf{i} - 3\mathbf{j} + \lambda(\mathbf{i} - 2\mathbf{j} + 5\mathbf{k})$. The point $A$ and the line $l$ lies in the plane $p_1$.

Show that the equation of $p_1$ is $2x + y = 1$.

The plane $p_2$ with equation $8x + ay + z = 4$ meets the plane $p_1$ at the line $L$ and the point $B(0, 1, 0)$ lies on $L$.

Show that $L$ is parallel to $\mathbf{i} - 2\mathbf{j} + (2a - 8)\mathbf{k}$.

Find the angle between $l$ and $p_2$.

Another point $C$ lies in the plane $p_1$ such that $BC$ is perpendicular to the line $L$.

Show that $\overrightarrow{BC}$ is parallel to $\mathbf{k}$. Given that the distance of $C$ from $p_2$ is 5 units, find the possible coordinates of $C$.

Perpendicular Projection and Reflection in a Line

Perpendicular Vectors Using the Scalar Product

<p>The planes $\pi_1$ and $\pi_2$ have equations $3x+c(y+z)-2=0$ and $\mathbf{r}=(\mathbf{i}+3\mathbf{j}-2\mathbf{k})+s(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+t(\mathbf{i}-\mathbf{k})$ respectively, where&nbsp;$c$ is a constant, and $s$ and $t$ are parameters. The point $A(1,3,-2)$ lies in both planes.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$c=-1$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Show that the vector equation of the line of intersection of $\pi_1$ and $\pi_2$, line $l$, is given by $\mathbf{r}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\alpha(\mathbf{i}-\mathbf{j}+4\mathbf{k})$, where&nbsp;$\alpha$ is a parameter.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the position vectors of the points on the line $l$ which are a distance of $3\sqrt{2}$ from the point $B(2,-3,7)$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Find the equation of the plane $\pi_3$ which is parallel to $\pi_2$ and contains the point $B$.</div><div class="list-item-paragraph">Hence show that the distance between the planes $\pi_2$ and $\pi_3$ is $\frac{20}{3\sqrt{3}}$.<span class="inline-marks">[3]</span></div></li></ol>

The planes $\pi_1$ and $\pi_2$ have equations $3x+c(y+z)-2=0$ and $\mathbf{r}=(\mathbf{i}+3\mathbf{j}-2\mathbf{k})+s(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+t(\mathbf{i}-\mathbf{k})$ respectively, where $c$ is a constant, and $s$ and $t$ are parameters. The point $A(1,3,-2)$ lies in both planes.

Magnitude of a Vector

2025 ACJC P1 Q9

<p>The line $l_1$ has the vector equation $\mathbf{r} = \begin{pmatrix} a \\ 1 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 0 \\ 1 \\ b \end{pmatrix}$, $\lambda \in \mathbb{R}$, where&nbsp;$a$ and $b$ are real constants.</p>
<p>The plane $p_1$ has cartesian equation $x + 2y + 4z = 4$.</p>
<p>The line $l_1$ is perpendicular to the line $l_2$ with cartesian equation $\frac{1-y}{2} = z - 3$, $x = 5$.</p>
<p>The line $l_1$ intersects the plane $p_1$ at the point with coordinates $(a, 1, 0)$.</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">By first finding the vector equation of line $l_2$, show that&nbsp;$a = b = 2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">Verify that the point $A$ with coordinates $(2, 2, 2)$ lies on $l_1$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Find the coordinates of the foot of perpendicular, $F$, from point $A$ to the plane $p_1$.<br>Hence find the shortest distance from point $A$ to the plane $p_1$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-iv"><div class="list-item-paragraph">The plane $p_2$ contains the line $l_1$ and the point $F$. Find the vector equation of the plane $p_2$, giving your answer in the form $\mathbf{r} \cdot \mathbf{n} = d$, where&nbsp;$\mathbf{n}$ is a vector and $d$ is a real constant to be determined.<span class="inline-marks">[2]</span></div></li></ol>

The line $l_1$ has the vector equation $\mathbf{r} = \begin{pmatrix} a \\ 1 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 0 \\ 1 \\ b \end{pmatrix}$, $\lambda \in \mathbb{R}$, where $a$ and $b$ are real constants.

The plane $p_1$ has cartesian equation $x + 2y + 4z = 4$.

The line $l_1$ is perpendicular to the line $l_2$ with cartesian equation $\frac{1-y}{2} = z - 3$, $x = 5$.

The line $l_1$ intersects the plane $p_1$ at the point with coordinates $(a, 1, 0)$.

Equations of Lines

2024 ACJC Promo Q11

<p>With respect to the origin $O$, the position vectors of the points $A$ and $B$ are $\mathbf{i}+2\mathbf{j}+\mathbf{k}$ and $\mathbf{i}-4\mathbf{k}$ respectively. The equation of the planes ${{p}_{1}}$ and ${{p}_{2}}$ are given by ${{p}_{1}}:\mathbf{r}\cdot \left( \begin{matrix}
   1  \\
   3  \\
   0  \\
\end{matrix} \right)=1$ and ${{p}_{2}}:\mathbf{r}\cdot \left( \begin{matrix}
   0  \\
   7  \\
   \alpha   \\
\end{matrix} \right)=-4\alpha $, where&nbsp;$\alpha $ is a real constant with $\alpha &gt;0$.<br></p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find the position vector of the foot of perpendicular from $A$ to $p_1$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Verify that&nbsp;$B$ lies on both&nbsp;$p_1$ and $p_2$. Show that the vector equation of the line of intersection, $L$, of $p_1$ and $p_2$ is of the form<span class="inline-marks">[2]</span></div></li></ol>
<p style="text-align: center">$\mathbf{r}=\left( \begin{matrix}
   1  \\
   0  \\
   -4  \\
\end{matrix} \right)+s\left( \begin{matrix}
   3\alpha   \\
   -\alpha   \\
   7  \\
\end{matrix} \right)$, $s\in \mathbb{R}$.<br><span class="inline-marks">[2]</span></p>
<ol class="i-list" start="3"><li data-list-item-id="item-iii" data-list-item-path="item-iii"><ol class="abc-list"><li data-list-item-id="item-iii-a" data-list-item-path="item-iii/item-iii-a"><div class="list-item-paragraph">The point $C(-1, 1, -4)$ is equidistant from both&nbsp;$p_1$ and $p_2$. Calculate the value of $\alpha$ for this case. Hence find in scalar product form the equation of the plane $p_3$ in which both the line $L$ and the point $C$ lie.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-iii-b" data-list-item-path="item-iii/item-iii-b"><div class="list-item-paragraph">The plane $p_4$ has equation $y - z = 2$. Given that the angle between $L$ and $p_4$ is $\frac{\pi}{6}$, find the value of $\alpha$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

With respect to the origin $O$, the position vectors of the points $A$ and $B$ are $\mathbf{i}+2\mathbf{j}+\mathbf{k}$ and $\mathbf{i}-4\mathbf{k}$ respectively. The equation of the planes ${{p}_{1}}$ and ${{p}_{2}}$ are given by ${{p}_{1}}:\mathbf{r}\cdot \left( \begin{matrix}
   1  \\
   3  \\
   0  \\
\end{matrix} \right)=1$ and ${{p}_{2}}:\mathbf{r}\cdot \left( \begin{matrix}
   0  \\
   7  \\
   \alpha   \\
\end{matrix} \right)=-4\alpha $, where $\alpha $ is a real constant with $\alpha >0$.


Find the position vector of the foot of perpendicular from $A$ to $p_1$.

Verify that $B$ lies on both $p_1$ and $p_2$. Show that the vector equation of the line of intersection, $L$, of $p_1$ and $p_2$ is of the form

$\mathbf{r}=\left( \begin{matrix}
   1  \\
   0  \\
   -4  \\
\end{matrix} \right)+s\left( \begin{matrix}
   3\alpha   \\
   -\alpha   \\
   7  \\
\end{matrix} \right)$, $s\in \mathbb{R}$.


2023 ASRJC Promo Q11

<p>Two planes $p_1$ and $p_2$ have respective cartesian equations given by</p>
<p style="text-align: center">\[4x + y + z = 8 \text{ and}\hspace{0.5em} 4x + 3y - z = 0.\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the sine of the acute angle between $p_1$ and $p_2$ in the form $\frac{m}{\sqrt{n}}$ where&nbsp;$m$ and $n$ are positive integers to be determined.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Verify that the point $A$ with coordinates $(1,0,4)$ lies on $p_1$ and $p_2$. Hence, without the use of a calculator, find the vector equation of the line $l$ formed by the intersection of $p_1$ and $p_2$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">It is given that&nbsp;$B$ is a point on $p_1$ with coordinates $(1,3,1)$. Show that&nbsp;$AB$ is perpendicular to $l$ and hence use <strong>(a)</strong> to deduce exactly the shortest distance from $B$ to $p_2$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the cartesian equation of the plane $p_3$ which contains $B$ and is perpendicular to both&nbsp;$p_1$ and $p_2$.</span><span class="inline-marks">[2]</span></div></li></ol>

Two planes $p_1$ and $p_2$ have respective cartesian equations given by

\[4x + y + z = 8 \text{ and } 4x + 3y - z = 0.\]

Find the sine of the acute angle between $p_1$ and $p_2$ in the form $\frac{m}{\sqrt{n}}$ where $m$ and $n$ are positive integers to be determined.

Verify that the point $A$ with coordinates $(1,0,4)$ lies on $p_1$ and $p_2$. Hence, without the use of a calculator, find the vector equation of the line $l$ formed by the intersection of $p_1$ and $p_2$.

Find the cartesian equation of the plane $p_3$ which contains $B$ and is perpendicular to both $p_1$ and $p_2$.

2025 NYJC P2 Q5

<p>The line $\ell$ has the equation $\frac{x-a}{2}=y-2=\frac{1-2z}{b}$, where&nbsp;$a$ and $b$ are real constants. The plane $\pi$ has the equation $3x-y+4z=10$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Given that the line $\ell$ and the plane $\pi$ do not intersect, show that&nbsp;$a \neq \frac{10}{3}$ and $b = \frac{5}{2}$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">It is given that&nbsp;$a=1$ and $b=3$. Find the point of intersection between the line $\ell$ and the plane $\pi$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">It is given now that&nbsp;$a=1, b=\frac{5}{2}$.</div><ol class="i-list"><li data-list-item-id="item-c-i" data-list-item-path="item-c/item-c-i"><div class="list-item-paragraph">Find the distance between the line $\ell$ and the plane $\pi$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c-ii" data-list-item-path="item-c/item-c-ii"><div class="list-item-paragraph">Determine whether the line $\ell$ and the origin $O$ lie on the same side of the plane $\pi$. State an equation of the other plane that is equidistant from line $\ell$ and parallel to plane $\pi$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

The line $\ell$ has the equation $\frac{x-a}{2}=y-2=\frac{1-2z}{b}$, where $a$ and $b$ are real constants. The plane $\pi$ has the equation $3x-y+4z=10$.

Given that the line $\ell$ and the plane $\pi$ do not intersect, show that $a \neq \frac{10}{3}$ and $b = \frac{5}{2}$.

It is given that $a=1$ and $b=3$. Find the point of intersection between the line $\ell$ and the plane $\pi$.

It is given now that $a=1, b=\frac{5}{2}$.

2025 ASRJC P1 Q9

<p>The planes $\pi_1$ and $\pi_2$ have equations $3x+c(y+z)-2=0$ and $\mathbf{r}=(\mathbf{i}+3\mathbf{j}-2\mathbf{k})+s(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+t(\mathbf{i}-\mathbf{k})$ respectively, where&nbsp;$c$ is a constant, and $s$ and $t$ are parameters. The point $A(1, 3, -2)$ lies in both planes.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$c=-1$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the vector equation of the line of intersection of $\pi_1$ and $\pi_2$, line $l$, is given by $\mathbf{r}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\alpha(\mathbf{i}-\mathbf{j}+4\mathbf{k})$, where&nbsp;$\alpha$ is a parameter.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the position vectors of the points on the line $l$ which are a distance of $3\sqrt{2}$ from the point $B(2, -3, 7)$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the equation of the plane $\pi_3$ which is parallel to $\pi_2$ and contains the point $B$. Hence show that the distance between the planes $\pi_2$ and $\pi_3$ is $\frac{20}{3\sqrt{3}}$.</span><span class="inline-marks">[3]</span></div></li></ol>

The planes $\pi_1$ and $\pi_2$ have equations $3x+c(y+z)-2=0$ and $\mathbf{r}=(\mathbf{i}+3\mathbf{j}-2\mathbf{k})+s(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+t(\mathbf{i}-\mathbf{k})$ respectively, where $c$ is a constant, and $s$ and $t$ are parameters. The point $A(1, 3, -2)$ lies in both planes.

Show that the vector equation of the line of intersection of $\pi_1$ and $\pi_2$, line $l$, is given by $\mathbf{r}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\alpha(\mathbf{i}-\mathbf{j}+4\mathbf{k})$, where $\alpha$ is a parameter.

Find the position vectors of the points on the line $l$ which are a distance of $3\sqrt{2}$ from the point $B(2, -3, 7)$.

Find the equation of the plane $\pi_3$ which is parallel to $\pi_2$ and contains the point $B$. Hence show that the distance between the planes $\pi_2$ and $\pi_3$ is $\frac{20}{3\sqrt{3}}$.

<p>A plane $\Pi$ has an equation of $\mathbf{r}\cdot\mathbf{n}=k$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">If the origin is on the plane explain how the value of $k$ could be determined and state its value.</div></li><li><div class="list-item-paragraph">State the unit vectors of the normal of $x-y$, $y-z$ and $x-z$ planes. Thus, find the equations of each plane.</div></li><li><div class="list-item-paragraph">Show that the shortest distance of plane $\Pi$ to the origin is $\frac{|k|}{|\mathbf{n}|}$.</div></li></ol>

A plane $\Pi$ has an equation of $\mathbf{r}\cdot\mathbf{n}=k$.

Unit Vectors

The x-y Plane, y-z Plane and x-z Plane Equation

<p>A plane $\Pi $ has an equation of $\mathbf{r}\cdot \mathbf{n}=k$.</p>
<ol class="i-list"><li data-list-item-id="item-1761379055736-nfi71mejm" data-list-item-path="item-1761379055736-nfi71mejm"><div class="list-item-paragraph">If the origin is on the plane explain how the value of $k$ could be determined and state its value.</div></li><li data-list-item-id="item-1761379056906-gt2h68qom" data-list-item-path="item-1761379056906-gt2h68qom"><div class="list-item-paragraph">State the unit vectors of the normal of $x-y$, $y-z$ and $x-z$ planes. Thus, find the equations of each plane.</div></li><li data-list-item-id="item-1761379057633-ye48f5lup" data-list-item-path="item-1761379057633-ye48f5lup"><div class="list-item-paragraph">Show that the shortest distance of plane $\Pi $ to the origin is $\frac{\left| k \right|}{\left| \mathbf{n} \right|}$.</div></li></ol>

A plane $\Pi $ has an equation of $\mathbf{r}\cdot \mathbf{n}=k$.

<p>The diagram (not drawn to scale) shows the structure of a partially constructed building that is built on a horizontal ground. The building has a square base foundation of $7$ m in length. Points $O$, $A$, $B$ and $C$ are the corners of the foundation of the building. The building currently consists of three vertical pillars $OD$, $AE$ and $CF$ of heights $8$ m, $5$ m and $5$ m respectively. A canvas is currently attached at $D$, $E$ and $F$, forming a temporary shelter for the building. $O$ is taken as the origin and vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$, each of length $1$ m, are taken along $\overrightarrow{OA}$, $\overrightarrow{OC}$ and $\overrightarrow{OD}$ respectively.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761377134885-p2eiy7gjncc.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzNzcxMzQ4ODUtcDJlaXk3Z2puY2MucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDg4MCwiZXhwIjoxNzkzNTA1MTgwfQ.3qnTo0YHn05NeLoDRTzh3fxSlrwS3EMQaBhC0oHqvZM" style="max-width: min(500px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-paragraph">Find a Cartesian equation of the plane that represents the canvas $DEF$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the acute angle which the canvas $DEF$ makes with the horizontal ground.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that the canvas is to be extended along the plane $DEF$ till it touches the horizontal ground, explain why point $B$ will lie beneath the canvas.<span class="inline-marks">[2]</span></div></li></ol>
<p>A cement roof is to be built to replace the extended canvas. A vertical partition wall is also to be built such that it is $d$ m away from and parallel to the plane $ODFC$, where&nbsp;$0 &lt; d &lt; 7$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Find the exact vector equation of the line where the roof meets the partition wall. Show your working clearly, leaving your answer in terms of $d$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">A lighting point, $P$, is to be placed on the roof such that it is closest to $B$. Find the position vector of $P$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram (not drawn to scale) shows the structure of a partially constructed building that is built on a horizontal ground. The building has a square base foundation of $7$ m in length. Points $O$, $A$, $B$ and $C$ are the corners of the foundation of the building. The building currently consists of three vertical pillars $OD$, $AE$ and $CF$ of heights $8$ m, $5$ m and $5$ m respectively. A canvas is currently attached at $D$, $E$ and $F$, forming a temporary shelter for the building. $O$ is taken as the origin and vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$, each of length $1$ m, are taken along $\overrightarrow{OA}$, $\overrightarrow{OC}$ and $\overrightarrow{OD}$ respectively.

A cement roof is to be built to replace the extended canvas. A vertical partition wall is also to be built such that it is $d$ m away from and parallel to the plane $ODFC$, where $0 < d < 7$.

Three-Dimensional Geometry

2017 NJC P2 Q4

<p>The equations of two planes${{p}_{1}}$ and ${{p}_{2}}$ are</p>
<p style="text-align: center">$2x+sy+z=-3$,</p>
<p style="text-align: center">$x-2y+3z=1$,</p>
<p>respectively, where&nbsp;$s$ is a negative real constant.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find a vector parallel to both&nbsp;${{p}_{1}}$ and ${{p}_{2}}$, giving your answer in terms of $s$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">It is given that&nbsp;${{p}_{1}}$ and ${{p}_{2}}$ intersect in a line $l$, and $l$ meets the $~xz$-plane at a point $A$. Find the coordinates of $A$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The shortest distance between the origin $O$ and ${{p}_{1}}$ is $\frac{3}{\sqrt{6}}$. Find the value of $s$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is now given that&nbsp;$s=-2$, with the normal vectors to ${{p}_{1}}$ and ${{p}_{2}}$ denoted as&nbsp;${{\mathbf{n}}_{1}}=\left( \begin{matrix}
2 \\
-2 \\
1 \\
\end{matrix} \right)$ and ${{\mathbf{n}}_{2}}=\left( \begin{matrix}
1 \\
-2 \\
3 \\
\end{matrix} \right)$ respectively. It is given that&nbsp;${{p}_{1}}$ and ${{p}_{2}}$ divide the real space into four regions ${{R}_{1}}$, , and as shown in the following two-dimensional diagram (not drawn to scale).</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761369246221-w64f672kic.jpg?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzNjkyNDYyMjEtdzY0ZjY3MmtpYy5qcGciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODgwLCJleHAiOjE3OTM1MDUxODB9.bWpXAMbSgHrX2ByS8l21SWqbhWsljFFDF9fThpBMKDU" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Relative to the origin $O$, the point $B$ has position vector $\mathbf{k}$. Determine whether $B$ lies in ${{R}_{1}}$, ${{R}_{2}}$, ${{R}_{3}}$ or ${{R}_{4}}$ , justifying your answer.<span class="inline-marks">[2]</span></div></li></ol>

The equations of two planes${{p}_{1}}$ and ${{p}_{2}}$ are

respectively, where $s$ is a negative real constant.

It is now given that $s=-2$, with the normal vectors to ${{p}_{1}}$ and ${{p}_{2}}$ denoted as ${{\mathbf{n}}_{1}}=\left( \begin{matrix}
2 \\
-2 \\
1 \\
\end{matrix} \right)$ and ${{\mathbf{n}}_{2}}=\left( \begin{matrix}
1 \\
-2 \\
3 \\
\end{matrix} \right)$ respectively. It is given that ${{p}_{1}}$ and ${{p}_{2}}$ divide the real space into four regions ${{R}_{1}}$, , and as shown in the following two-dimensional diagram (not drawn to scale).

Finding Points on a Plane

Vector (Cross) Product

2023 HCI P2 Q5

<ol class="abc-list"><li data-list-item-id="item-1761366698278-e7ksc9uy0" data-list-item-path="item-1761366698278-e7ksc9uy0"><div class="list-item-paragraph">Give a geometrical interpretation of what it means when</div><ol class="i-list"><li data-list-item-id="subitem-1761366701031-bmjcxfrlh" data-list-item-path="item-1761366698278-e7ksc9uy0/subitem-1761366701031-bmjcxfrlh"><div class="list-item-paragraph">solving the equations $\mathbf{r}=\mathbf{a}+t\mathbf{b}$ and $\mathbf{r}\cdot \mathbf{n}=k$ gives infinitely many solutions, where&nbsp;$\mathbf{a}$, $\mathbf{b}$ and $\mathbf{n}$ are constant vectors, $t$ is a real parameter and $k$ is a constant scalar.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1761366701946-g8uv14rle" data-list-item-path="item-1761366698278-e7ksc9uy0/subitem-1761366701946-g8uv14rle"><div class="list-item-paragraph">solving the equations $\mathbf{r}=\mathbf{a}+t\mathbf{b}$ and $\mathbf{r}=\mathbf{c}+m\mathbf{d}$ does not give a unique solution, where&nbsp;$\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ and $\mathbf{d}$ are constant vectors, $t$ and $m$ are real parameters.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="item-1761366699193-ild3j7jc2" data-list-item-path="item-1761366699193-ild3j7jc2"><div class="list-item-paragraph">A plane $p$ contains the point $A$ with position vector $3\mathbf{i}+9\mathbf{k}$ and is perpendicular to vector $-4\mathbf{i}+5\mathbf{j}+3\mathbf{k}$.</div><ol class="i-list"><li data-list-item-id="subitem-1761366704474-g1qcyak4c" data-list-item-path="item-1761366699193-ild3j7jc2/subitem-1761366704474-g1qcyak4c"><div class="list-item-paragraph">Find the exact shortest distance from the origin $O$ to plane $p$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1761366705028-evq8rs72k" data-list-item-path="item-1761366699193-ild3j7jc2/subitem-1761366705028-evq8rs72k"><div class="list-item-paragraph">By considering the plane containing the point $D$ with coordinates $\left( 3,-2,4 \right)$ and parallel to $p$ or otherwise, determine whether the origin $O$ and $D$ are on the same or opposite side of $p$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1761366705868-gihyhczz2" data-list-item-path="item-1761366699193-ild3j7jc2/subitem-1761366705868-gihyhczz2"><div class="list-item-paragraph">Show that the coordinates of the foot of the perpendicular from $D$ to $p$ is $\left( 1,\frac{1}{2},\frac{11}{2} \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1761366706536-clk8mzwmy" data-list-item-path="item-1761366699193-ild3j7jc2/subitem-1761366706536-clk8mzwmy"><div class="list-item-paragraph">Hence find an equation that describes the set of all points which are of the same distance from $p$ as&nbsp;$D$ but on the opposite side of $p$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Relationships Between Two Lines

2020 TMJC P1 Q7

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761365257856-bwiph4eca4.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzNjUyNTc4NTYtYndpcGg0ZWNhNC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODgwLCJleHAiOjE3OTM1MDUxODB9.cWA85SvcKmiLxoiR8m-7BYxEPFwSN9uwT5Xa5ty0j-Q" style="max-width: 300px; height: auto; display: block;" class="aligncenter" /></figure>
<p>The diagram above shows part of the structure of a rock-climbing wall, with a horizontal base $OABC$, a vertical rectangular wall $OADE$ and an inclined rectangular wall $DEFG$. With respect to the origin $O$, the vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ each of length $1$ m, are parallel to $OA$, $OC$ and $OE$ respectively. It is given that $OE=10$m.</p>
<p>The plane $OADE$ has equation $\mathbf{r}\cdot \left( \begin{matrix}
0 \\
1 \\
0 \\
\end{matrix} \right)=p$, and the plane $DEFG$ has equation $\mathbf{r}=\left( \begin{matrix}
0 \\
0 \\
10 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
3 \\
-2 \\
4 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
0 \\
-1 \\
q \\
\end{matrix} \right)$, where $p$ and $q$ are constants, $\lambda $ and $\mu $ are parameters.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Give a reason why $p=0$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Write down a vector equation of the line passing through the points $D$ and $E$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Show that $q=2$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence, find the obtuse angle between the walls $OADE$ and $DEFG$.<span class="inline-marks">[3]</span></div></li></ol>
<p>A rock climber is at the point $H\left( 2,\,-1,\,12 \right)$.</p>
<ol class="abc-list" start="5"><li><div class="list-item-paragraph">Show that the climber is on the plane $DEFG$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The line $l$ passing through the points $F$ and $G$ has equation $\mathbf{r}=\left( \begin{matrix}
3 \\
-2 \\
14 \\
\end{matrix} \right)+\gamma \left( \begin{matrix}
1 \\
0 \\
0 \\
\end{matrix} \right)$, where $\gamma $ is a parameter.</p>
<ol class="abc-list" start="6"><li><div class="list-item-paragraph">Find the exact shortest distance from the climber to $l$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram above shows part of the structure of a rock-climbing wall, with a horizontal base $OABC$, a vertical rectangular wall $OADE$ and an inclined rectangular wall $DEFG$. With respect to the origin $O$, the vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ each of length $1$ m, are parallel to $OA$, $OC$ and $OE$ respectively. It is given that $OE=10$m.

The plane $OADE$ has equation $\mathbf{r}\cdot \left( \begin{matrix}
0 \\
1 \\
0 \\
\end{matrix} \right)=p$, and the plane $DEFG$ has equation $\mathbf{r}=\left( \begin{matrix}
0 \\
0 \\
10 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
3 \\
-2 \\
4 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
0 \\
-1 \\
q \\
\end{matrix} \right)$, where $p$ and $q$ are constants, $\lambda $ and $\mu $ are parameters.

A rock climber is at the point $H\left( 2,\,-1,\,12 \right)$.

The line $l$ passing through the points $F$ and $G$ has equation $\mathbf{r}=\left( \begin{matrix}
3 \\
-2 \\
14 \\
\end{matrix} \right)+\gamma \left( \begin{matrix}
1 \\
0 \\
0 \\
\end{matrix} \right)$, where $\gamma $ is a parameter.

2022 YIJC Promo Q11

<p>The coordinates of the points $A$, $B$ and $C$ are given by $A\,(0,\,2,\,4)$, $B\,(4,\,6,\,11)$ and $C\,(8,\,1,\,0)$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the triangle with vertices $A$, $B$ and $C$ is an isosceles right-angled triangle.</div></li><li><div class="list-item-paragraph">Find the point $D$ in the same plane as $A$, $B$ and $C$ such that $BCD$ is an equilateral triangle.</div></li></ol>

The coordinates of the points $A$, $B$ and $C$ are given by $A\,(0,\,2,\,4)$, $B\,(4,\,6,\,11)$ and $C\,(8,\,1,\,0)$.

Finding a point to form an equilateral triangle given a line segment in 3D vectors

<p>The plane $p$ passes through the points $A$, $B$ and $C$ with coordinates $\left( -3,-4,11 \right)$, $\left( 1,-2,0 \right)$ and $\left( -5,2,-1 \right)$ respectively. The point $M$ has position vector given by $\mathbf{m}=3\mathbf{i}+6\mathbf{j}+15\mathbf{k}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $\overrightarrow{AM}$ is perpendicular to $p$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the coordinates of $N$ which is the mirror image of $M$ in $p$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find a vector equation of the line which is a reflection of the line $MB$ in $p$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The plane $q$ has equation $11x+6y-z-4+k\left( -x-3y+z-4 \right)=0$ for some constant $k$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Given that $q$ contains $M$, find a cartesian equation of $q$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">By finding the line of intersection between $p$ and $q$, or otherwise, find a cartesian equation of the plane which is a reflection of $q$ in $p$.</div><p style="text-align: right;">[5]</p></li></ol>

The plane $p$ passes through the points $A$, $B$ and $C$ with coordinates $\left( -3,-4,11 \right)$, $\left( 1,-2,0 \right)$ and $\left( -5,2,-1 \right)$ respectively. The point $M$ has position vector given by $\mathbf{m}=3\mathbf{i}+6\mathbf{j}+15\mathbf{k}$.

The plane $q$ has equation $11x+6y-z-4+k\left( -x-3y+z-4 \right)=0$ for some constant $k$.

2019 NJC Promo Q11

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761310225100-tlva88td4hc.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzMTAyMjUxMDAtdGx2YTg4dGQ0aGMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDg4MCwiZXhwIjoxNzkzNTA1MTgwfQ.ovBOzVtr311InjQZxJgaGjO8qIhPxSxbJRTzBRVHELs" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>A light ray passes from air to a vertical reflective panel, ${{p}_{1}}$ (see diagram). The reflective panel, ${{p}_{1}}$ is part of the surface which has the equation $ax+y-z=3$, where&nbsp;$a$ is a constant. Referenced to an origin $O$, the light ray passes through point $A\left( 2,1,4 \right)$ and hits the panel at point $B\left( 4,1,2 \right)$. It then gets reflected to form another light ray along $BC$. The path of the light ray and its reflected path make the same angle $\theta $ with ${{p}_{1}}$.</p>
<ol class="i-list"><li data-list-item-id="item-1761310204215-03u4b6qvd" data-list-item-path="item-1761310204215-03u4b6qvd"><div class="list-item-paragraph">Show that&nbsp;$a=1$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1761310207724-vkbjd6832" data-list-item-path="item-1761310207724-vkbjd6832"><div class="list-item-paragraph">Find the projection vector of the light ray along $AB$ onto ${{p}_{1}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761310208325-v93i8hg34" data-list-item-path="item-1761310208325-v93i8hg34"><div class="list-item-paragraph">Find a cartesian equation of the reflected light ray along $BC$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1761310208925-57tjrilcn" data-list-item-path="item-1761310208925-57tjrilcn"><div class="list-item-paragraph">Two other distinct panels parallel to ${{p}_{1}}$ are each placed at a distance of $\frac{10}{\sqrt{3}}$ units away from ${{p}_{1}}$. Find the vector equations of these panels.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761310210095-fpk1emj59" data-list-item-path="item-1761310210095-fpk1emj59"><div class="list-item-paragraph">The light ray $BC$ hits one of the panels. Determine which is the panel and provide a brief reason.<span class="inline-marks">[1]</span></div></li></ol>

A light ray passes from air to a vertical reflective panel, ${{p}_{1}}$ (see diagram). The reflective panel, ${{p}_{1}}$ is part of the surface which has the equation $ax+y-z=3$, where $a$ is a constant. Referenced to an origin $O$, the light ray passes through point $A\left( 2,1,4 \right)$ and hits the panel at point $B\left( 4,1,2 \right)$. It then gets reflected to form another light ray along $BC$. The path of the light ray and its reflected path make the same angle $\theta $ with ${{p}_{1}}$.

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